{"paper":{"title":"New Algorithms for Parity-SAT and Its Bounded-Occurrence Versions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Randomized algorithms solve Parity-d-occ-SAT in O^*(2^{m(1-1/O(d))}) time for any fixed d.","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Frank Stephan, Haoyun Tang, Junqiang Peng, Mingyu Xiao, Sanjay Jain","submitted_at":"2026-05-13T20:25:11Z","abstract_excerpt":"Parity-SAT is the problem of determining whether a given CNF formula has an odd number of satisfying assignments. As a canonical $\\oplus$P-complete problem, it represents a fundamental variant of the exact model counting problem (#SAT). Under the Strong Exponential Time Hypothesis (SETH), Parity-SAT admits no $O^*((2-\\varepsilon)^n)$-time or $O^*((2-\\varepsilon)^m)$-time algorithm for any constant $\\varepsilon>0$, where $n$ and $m$ denote the numbers of variables and clauses, respectively. Thus, breaking the $2^n$ or $2^m$ barrier appears impossible in full generality.\n  In this work, we revis"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We design a randomized O^*(2^{m(1-1/O(d))})-time algorithm for Parity-d-occ-SAT, thereby breaking the 2^m barrier for every fixed d. For d=2 we obtain O^*(1.1193^n) or O^*(1.3248^m), and for general Parity-SAT an O^*(1.1052^L) algorithm.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The algorithms assume that parity can be exploited via structural reductions and branching rules that avoid the overhead of exact counting, without hidden costs that would appear only in a full proof or implementation.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"New randomized and branching algorithms achieve O*(2^{m(1-1/O(d))}) time for Parity-d-occ-SAT and O*(1.1052^L) time for general Parity-SAT, outperforming exact counting bounds.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Randomized algorithms solve Parity-d-occ-SAT in O^*(2^{m(1-1/O(d))}) time for any fixed d.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5b6e6683f75ce51e0a14a6698aca48d6cb89be388ef6ef94fea00260a34a7c85"},"source":{"id":"2605.14093","kind":"arxiv","version":1},"verdict":{"id":"8d4cd976-1403-468b-9aed-5ca4d8674921","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:02:37.540963Z","strongest_claim":"We design a randomized O^*(2^{m(1-1/O(d))})-time algorithm for Parity-d-occ-SAT, thereby breaking the 2^m barrier for every fixed d. For d=2 we obtain O^*(1.1193^n) or O^*(1.3248^m), and for general Parity-SAT an O^*(1.1052^L) algorithm.","one_line_summary":"New randomized and branching algorithms achieve O*(2^{m(1-1/O(d))}) time for Parity-d-occ-SAT and O*(1.1052^L) time for general Parity-SAT, outperforming exact counting bounds.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The algorithms assume that parity can be exploited via structural reductions and branching rules that avoid the overhead of exact counting, without hidden costs that would appear only in a full proof or implementation.","pith_extraction_headline":"Randomized algorithms solve Parity-d-occ-SAT in O^*(2^{m(1-1/O(d))}) time for any fixed d."},"references":{"count":47,"sample":[{"doi":"","year":1971,"title":"Cook , title =","work_id":"7a1ba4b8-f6e6-4578-b28c-2fac8ba3c03f","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1979,"title":"Valiant , title =","work_id":"1aebe9ac-f4b9-4d17-982b-a3e211e28ecf","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2003,"title":"Proceedings of the 44th Annual Symposium on Foundations of Computer Science (","work_id":"0742f591-00ad-4b13-a099-94779134da30","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1996,"title":"Dan Roth , title =. Artif. Intell. , volume =. 1996 , doi =","work_id":"0df74d66-3e8f-4685-9ec6-6fbfce3b1221","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Kautz , title =","work_id":"662a91d3-ed7a-465d-8746-dcd0d31322fb","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":47,"snapshot_sha256":"7f09ead516086669e4080a6685a242087eddc40410b11be115c25b403c03f248","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}